Quantitative Methods Seminar

Abstract: Diffusion models have become a cornerstone of modern generative AI, achieving state-of-the-art performance across image, audio, and video generation tasks. However, their effectiveness is highly sensitive to numerous design choices and hyperparameters. This talk will provide a self-contained introduction to diffusion models, followed by a principled framework for selecting one of their most crucial components: the noise schedule. The material will be accessible to a broad audience, with no prior knowledge of diffusion models required. This talk is based on ongoing work with Baris Ata (University of Chicago) and Harsha Honnappa (Purdue).

Abstract: Robust reinforcement learning (RL) focuses on designing optimal policies from data for MDPs with model uncertainties. Existing convergence guarantees for robust RL are either limited to tabular settings or use restrictive assumptions in the function approximation setting. We will present an RL algorithm for learning the optimal policy from data in the function approximation setting and provide finite‑time sample‑complexity bounds without requiring generative access to the underlying MDP model. Our algorithm uses a combination of ideas from distributionally robust optimization (DPO), two time-scale stochastic approximation, and traditional (non-robust) fitted value iteration and Q-learning.

Bio: R. Srikant is a Grainger Distinguished Chair in Engineering, and Professor in the Department of Electrical and Computer Engineering and the Coordinated Science Lab. His research interests include machine learning, applied probability, stochastic control, and communication networks. He is the recipient of the 2015 INFOCOM Achievement Award, the 2019 IEEE Koji Kobayashi Computers and Communications Award and the 2021 ACM SIGMETRICS Achievement Award. He has also received several Best Paper awards including the 2015 INFOCOM Best Paper Award, the 2017 Applied Probability Society Best Publication Award, and the 2017 WiOpt Best Paper award. He was the Editor-in-Chief of the IEEE/ACM Transactions on Networking from 2013-2017 and is currently an Area Editor for the Mathematics of Operations Research.

Abstract: This talk will present a new conceptual view of acceleration in smooth convex optimization. We will recover and motivate two landmark algorithms in smooth convex optimization, including Nesterov's Fast Gradient Method. Then, using variations of the same concept, we will derive two new accelerated methods: a (conjectured) minimax optimal accelerated gradient descent scheme without momentum, as well as an accelerated first-order method with convergence guarantees even stronger than minimax optimality.

This talk is based on joint work with Benjamin Grimmer and Kevin Shu.

Abstract: This talk presents recent work on learning the dynamics of complex systems modeled by high-dimensional Markov-switching ordinary differential equations (ODEs), a setting where the system evolves according to nonlinear additive ODEs with parameters switching over time based on a latent Markov process. Such models are highly relevant in fields like biology and neuroscience, where direct experimentation is often infeasible, and inference must rely on discrete-time observations. We introduce a two-stage estimation procedure that first reconstructs continuous sample paths from discrete data and then recovers model parameters. The method comes with linear convergence guarantees and sharp bounds on the statistical error, enabled by novel analysis techniques under mixing conditions via truncated posterior processes. As an application, we analyze resting-state brain activity data, revealing distinct switching behavior in the transition rate matrices between individuals with ADHD and typical controls. This is joint work with Katherine Tsai and Sanmi Koyejo, and is based on our preprint: High-Dimensional Markov-Switching Ordinary Differential Processes (arXiv:2501.00087).

Bio: Mladen Kolar is a professor in the Department of Data Sciences and Operations at the USC Marshall School of Business and a visiting professor in the Department of Statistics and Data Science at Mohamed bin Zayed University of Artificial Intelligence. Mladen earned his PhD in Machine Learning from Carnegie Mellon University in 2013. His research focuses on high-dimensional statistical methods, probabilistic graphical models, and scalable optimization methods, driven by the need to uncover interesting and scientifically meaningful structures from observational data. Mladen was selected as a recipient of the 2024 Junior Leo Breiman Award for his outstanding contributions to these areas. He currently serves as an associate editor for the Journal of Machine Learning Research and the Journal of Computational and Graphical Statistics. He also served as an associate editor for the Annals of Statistics.

Abstract: Solving real-world stochastic optimization problems presents two key challenges: the messiness of real-world data, which can be noisy, biased, or corrupted due to factors like heavy-tailed noise distributions, outliers, distribution shifts, and even adversarial attacks; and the laborious, time-intensive requirement of manually tuning step sizes in many existing algorithms. I will introduce a simple adaptive optimization framework that avoids the need for manual step size tuning by adaptively adjusting the step size in each iteration based on the progress of the algorithm. To model the highly noisy or corrupted input data, the algorithm only assumes access to function-related information through probabilistic oracles. Crucially, these oracles can return function value and gradient estimates contaminated with noise that is heavy tailed. This framework is very general, encompassing a wide range of algorithms. It is applicable to multiple problem settings, such as expected loss minimization in machine learning, simulation optimization, and derivative-free optimization. Under reasonable conditions on the oracles, we provide an analytical framework that bounds the iteration complexity with high probability for settings with highly noisy or corrupted inputs.

Abstract: Reinforcement learning from human feedback (RLHF) has emerged as the leading approach to aligning large language models (LLMs) with human preferences. Despite its success, two challenges remain fundamental: feedback is costly and heterogeneous across annotators, and the resulting reward models often lack principled measures of uncertainty. This talk presents recent advances that address these challenges by integrating tools from optimal design and statistical inference into the RLHF framework. First, I introduce a dual active learning approach, inspired by optimal design, that adaptively selects both conversations and annotators to maximize information gain, improving the efficiency of limited feedback budgets. Second, I present a framework for uncertainty quantification in reward learning, enabling valid statistical comparisons across LLM models and more reliable best-of-n alignment policies. Together, these results illustrate how statistics can help trustworthy and data-efficient LLM alignment.

Abstract: Effective supply chains are built not only on the movement of goods but also on the movement of information. At their core, inventory policies do more than manage stock; they govern how demand signals are transmitted, interpreted, and forecasted across firms. This talk explores how viewing replenishment decisions through the lens of forecasting and information design reshapes our understanding of coordination in decentralized supply chains. By applying tools from signal processing and harmonic analysis, we formulate and solve the policy design problem as an infinite-dimensional nonconvex optimization problem. We further show that the value of information lies less in direct data sharing and more in how orders embed signals that can be recovered upstream. This perspective yields two key contributions: a novel forecast-adjusted bullwhip measure and the design of simple, near-optimal ordering policies. Ultimately, this work clarifies when firms should share information, when they can rely on forecasting alone, and how the strategic use of delays or smoothing can both stabilize operations and mitigate the bullwhip effect. This presentation draws from https://doi.org/10.1287/moor.2023.0008, https://dx.doi.org/10.2139/ssrn.5077356, and related working papers.

Abstract: Many challenging problems in statistics and applied probability can be posed as that of minimizing a smooth functional over a linearly constrained space of probability measures supported on a compact subset of the Euclidean space. Examples include the classical experimental design problem, the P-means problem, and the problem of moments.  In this talk, I propose Frank-Wolfe (FW) type first-order recursions that operate on probability spaces. Two observations about the proposed recursions are salient. First, the influence function of the objective naturally emerges as the variational object. Second, the (infinite-dimensional) FW sub-problems are solved by a probability measure that concentrates on the minima of the influence function, leading to FW recursions that are simply stated, and often easily implemented. Incorporating constraints using an interior point framework, a derivative-free analogue, and a stochastic variation, all follow in a somewhat seamless fashion. To promote intuition, I will discuss illustrative examples with exact influence function calculations. I will also provide commentary on when such problems might be solved efficiently using the proposed framework. This is joint work with Di Yu (Purdue Statistics), Shane Henderson (Cornell ORIE), and Roelof Coetzer (North-West University).

Bio: Raghu Pasupathy is a Professor of Statistics at Purdue University. Prior to joining Purdue in 2014, Pasupathy spent nine years in the Industrial and Systems Engineering Department at Virginia Tech, first as an assistant professor and then as an associate professor. Pasupathy’s research interests lie in the theoretical and computational aspects of stochastic optimization, Monte Carlo, and uncertainty quantification. Pasupathy has been associated with the simulation and optimization communities in various capacities over the previous two decades. More information, including downloadable papers, can be obtained through his website at https://web.ics.purdue.edu/~pasupath/

Abstract: Understanding the inner workings of human brains, and their connections to both neurological disorders and normal development, is one of the most intriguing scientific questions. Advances in neuroscience have been greatly facilitated by various neuroimaging technologies, including anatomical magnetic resonance imaging (MRI), functional magnetic resonance imaging (fMRI), electroencephalography (EEG), diffusion tensor imaging, positron emission tomography (PET), among many others. The sheer size and complexity of imaging data, however, present significant challenges and call for continual development of new statistical methodologies. In this talk, I will provide an overview of a range of neuroscience motivated topics our group has been investigating, including imaging tensor analysis, brain connectivity network analysis, imaging causal analysis, and more. I will also illustrate with a number of specific case studies. Meanwhile, the methodologies developed are applicable to many other domains as well.

Bio: Lexin Li, Ph.D., is Professor of Biostatistics, and Divisional Chair, at the Department of Biostatistics and Epidemiology, and also Department of Statistics and Helen Wills Neuroscience Institute, at the University of California, Berkeley. His research interests include neuroimaging analysis, brain-computer-interface, deep and reinforcement learning, functional and point process data analysis, tensor data analysis, and network data analysis. He is a Fellow of the American Association for the Advancement of Science (AAAS), the Institute of Mathematical Statistics (IMS), the American Statistical Association (ASA), the Asia-Pacific Artificial Intelligence Association (AAIA), and an Elected Member of the International Statistical Institute (ISI). He is the Editor-in-Chief of the Annals of Applied Statistics for 2025-27.

Abstract: Low Earth Orbit (LEO) has become the most valuable real estate in space. Driven largely by the deployment of satellite mega-constellations (e.g., Starlink), the number of active satellites has increased by approximately 175% between 2020 and 2025. The rapid expansion of these constellations, combined with ambitious proposals for future launches, risks precipitating a tragedy of the commons in space. A particularly concerning outcome is the onset of cascading collisions - the Kessler Syndrome - which could generate vast debris clouds and render space operations increasingly hazardous, if not altogether impossible. These developments underscore the urgent need to understand the effective capacity of LEO. 

This talk presents ongoing research in the Stochastic Systems Lab at Purdue University that addresses this question through compartmentalized stochastic models. We establish fluid and diffusion approximations in high-density regimes and illustrate their behavior through simulations and fixed-point analyses to project the dynamics of LEO over multiple time-scales. Time permitting, I will also discuss our ongoing work on using local weak convergence methods to analyze the compartmental models, which offers a more nuanced approximation of these complex stochastic systems.

Bio: Harsha Honnappa is an Associate Professor in the Edwardson School of Industrial Engineering at Purdue University, where he directs the Stochastic Systems Lab. He is an applied probabilist with interests in the modeling and analysis of stochastic systems, theoretical statistics, stochastic optimization and control. His research is supported by multiple grants from the National Science Foundation, including an NSF CAREER award, the Office of Naval Research, the Purdue Research Foundation, and through the Edwardson School of Industrial Engineerings Frontiers awards. He is currently Area Editor for Stochastic Models for Operations Research Letters, and Associate Editor at Operations Research and Queueing Systems Journals.

Abstract: Reinforcement learning (RL) has become an increasingly popular framework for sequential decision-making, driven by milestone successes in game-playing AI, robotics, and recent large language models. These advances have, in turn, motivated theoretical studies aimed at guiding practical implementations. However, most existing theoretical results focus on episodic or discounted-reward settings, limiting their applicability to real-world operations research problems that demand continual, long-horizon decision-making, such as inventory management, dynamic pricing, supply chain control, and queueing networks. In this work, we study RL in the long-term average-reward setting and present the first finite-time last-iterate convergence analysis of asynchronous Q-learning. A key feature of the algorithm is its adaptive stepsize, which serves as a local clock for each state-action pair and implicitly performs importance sampling to mitigate the distribution shift caused by asynchronous updates. Technically, the use of adaptive stepsizes makes each update depend on the full sample history, rendering the algorithm a non-Markovian stochastic approximation (SA) process. To address this challenge, we develop (1) a time-inhomogeneous Markovian reformulation of non-Markovian SA, and (2) a framework that combines almost-sure time-varying bounds, conditioning arguments, and Markov chain concentration inequalities to decouple the dependencies between adaptive stepsizes and iterates. The resulting tools are broadly applicable to the analysis of general SA algorithms with adaptive stepsizes.

Bio: Dr. Zaiwei Chen is an Assistant Professor in the School of Industrial Engineering at Purdue University. He received his Ph.D. in Industrial and Systems Engineering from the Georgia Institute of Technology and completed his postdoctoral training at the California Institute of Technology. His research focuses on developing a foundational understanding of the design and analysis of stochastic iterative algorithms, with applications in reinforcement learning, learning in games, optimization, and control.

Abstract: Transformers have demonstrated remarkable chain-of-thought reasoning capabilities, yet, the underlying mechanisms by which they acquire and extrapolate these capabilities remain limited. This talk presents a theoretical analysis of transformers trained via gradient descent for symbolic reasoning and state tracking tasks with increasing problem complexity. Our analysis reveals the coordination of multi-head attention to solve multiple subtasks in a single autoregressive path, and the bootstrapping of inherently sequential reasoning through recursive self-training curriculum. Our optimization-based guarantees demonstrate that even shallow multi-head transformers, with chain-of-thought, can be trained to effectively solve problems that would otherwise require deeper architectures.

Bio: Dr. Yuejie Chi is the Charles C. and Dorothea S. Dilley Professor of Statistics and Data Science at Yale University, with a secondary appointment in Computer Science. She received her Ph.D. and M.A. from Princeton University, and B. Eng. (Hon.) from Tsinghua University, all in Electrical Engineering. Her research interests lie in the theoretical and algorithmic foundations of data science, generative AI, reinforcement learning, and signal processing, motivated by applications in scientific and engineering domains. Among others, Dr. Chi received the Presidential Early Career Award for Scientists and Engineers (PECASE), SIAM Activity Group on Imaging Science Best Paper Prize, IEEE Signal Processing Society Young Author Best Paper Award, and the inaugural IEEE Signal Processing Society Early Career Technical Achievement Award for contributions to high-dimensional structured signal processing. She is an IEEE Fellow (Class of 2023) for contributions to statistical signal processing with low-dimensional structures.

Abstract: Alignment plays a critical role in training high-quality Large Language Models (LLMs). It guides LLMs to generate content that matches human values and goals, and also serves as a safeguard mechanism to prevent LLMs from being exploited by malicious users to create inappropriate responses or spread misinformation. We present a theoretical framework showing that popular LLM alignment methods—including RLHF and its variants—can be viewed as divergence estimators between aligned (safe or preferred) and unaligned (harmful or less-preferred) distributions. This perspective explains the separation phenomenon in the latent space of aligned LLMs between safe and harmful prompts. This general divergence framework inspires a novel KL divergence-based alignment method (KLDO), whose empirical effectiveness is validated via simulation. We further demonstrate that using compliance–refusal datasets, rather than standard preference-based datasets, yields stronger separation and improved safety alignment. Finally, to quantify the separation effect, we propose a distance-based metric in the prompt representation space, which also acts as a statistically significant indicator for model safety.